1 Answer
1

Specifically addressing this question, let us work on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ filtered with $(\mathcal{F}_t)_{t \geq 0}$. We assume that the asset process under the physical measure $\mathbb{P}$ is:
$$\text{d}S_t=\alpha(t,S_t)\text{d}t+\sigma\text{d}W_t$$

where $\alpha(t,S_t)$ is an appropriate$^{*}$ process adapted to the filtration $(\mathcal{F}_t)_{t \geq 0}$.

Thus we need to change the drift of $S_t$ from $\alpha(t,S_t)$ to $rS_t$. Now let us define the following process:
$$ \tilde{W}_t=W_t+\int_0^t\frac{\alpha(t,S_t)-rS_t}{\sigma}\text{d}t$$

The process $\tilde{W}_t$ is a Brownian Motion under the risk-neutral measure $\mathbb{Q}$ defined by the Doléans-Dade exponential of the adapted process $f(t)=(\alpha(t,S_t)-rS_t)/\sigma$. Then we can make the change of measure$^{\text{*}}$:
$$\begin{align}
\text{d}S_t & = \alpha(t,S_t)\text{d}t+\sigma\text{d}W_t
\\[3pt]
& = \alpha(t,S_t)\text{d}t+\sigma\left(\text{d}\tilde{W}_t-\frac{\alpha(t,S_t)-rS_t}{\sigma}\text{d}t\right)
\\[3pt]
& = rS_t\text{d}t+\sigma\text{d}\tilde{W}_t
\end{align}$$

[Edit: why does $\text{Eq. 1}$ imply that the drift of the asset needs to be $rS_t\text{d}t$ under measure $\mathbb{Q}$? We know from risk-neutral theory that the price of a derivative is the expectation of its payoff under measure $\mathbb{Q}$ (see e.g. Proposition 2.9 from $[1]$); we therefore need the distribution of the asset under the risk-neutral measure; we also know that the discounted price of the asset, $X_t=D_tS_t$, is a martingale under $\mathbb{Q}$ (see e.g. Proposition 2.8 from $[1]$). On the other hand, (local) martingales can be characterised as a process without drift, i.e. for a martingale $M_t$:
$$M_t=\sigma(t,M_t)\text{d}W_t$$
for some adapted process $\sigma(t,M_t)$, which can be a constant $\sigma$. Thus given $\text{Eq. 1}$, which gives the SDE of $X_t$, in order to make the discounted asset a (local) martingale we need to cancel its drift under $\mathbb{Q}$ by making the drift of the asset $S_t$ to be equal to $rS_t\text{d}t$ under that same measure. In practice, we always work with local martingales that are also martingales hence we do not bother to make all the technical checks. See the technical note from my answer to "Why discounted derivative price is a martingale?" for more details.]

$\text{*:}$ we would need to check Novikov's condition to ensure the change of measure is legit, we assume process $\alpha(t,S_t)$ is such that the condition is fulfilled.

$\begingroup$Actually I knew most of your (really good) answer. At some point u write:"we need to change the drift to rSdt". This is where I struggle. Maybe because I haven't understood Risk Neutral theory totally. Can you add a couple of word for WHY we need to change the drift to exactly " $rS_t dt$"$\endgroup$
– SanjayMar 20 '18 at 21:32